A secret positive integer is assigned to each vertex of atriangle. On each side

Question

A secret positive integer is assigned to each vertex of atriangle. On each side of the triangle is written the sum of thesecret numbers at its ends. For example, we might have a trianglewith sides labeled 11, 18, and 27. What are the secret numbers fromwhich these numbers are derived? Find a simple rule for revealingthe secret numbers. What triples of numbers are possible for labelsfor the sides? What are impossible? Does it make a difference thatthe integers are positive? Why or why not? Generalize this questionto other polygons. Please show work. A secret positive integer is assigned to each vertex of atriangle. On each side of the triangle is written the sum of thesecret numbers at its ends. For example, we might have a trianglewith sides labeled 11, 18, and 27. What are the secret numbers fromwhich these numbers are derived? Find a simple rule for revealingthe secret numbers. What triples of numbers are possible for labelsfor the sides? What are impossible? Does it make a difference thatthe integers are positive? Why or why not? Generalize this questionto other polygons. Please show work.

Explanation / Answer

For the first question, if you let the three vertices be x, y,z, then you can set up a system of equations as follows: x + y = 11 x + z = 18 y + z = 27. Solving the system, we obtain the solutions x = 1, y = 10, z =17 (verify). This problem is interesting and I will try to work on theother questions later.

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